\(\int (d+e x^2) (a+c x^4) \, dx\) [123]

Optimal result

Integrand size = 15, antiderivative size = 32 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \]

[Out]

a*d*x+1/3*a*e*x^3+1/5*c*d*x^5+1/7*c*e*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1168} \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \]

[In]

Int[(d + e*x^2)*(a + c*x^4),x]

[Out]

a*d*x + (a*e*x^3)/3 + (c*d*x^5)/5 + (c*e*x^7)/7

Rule 1168

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps \begin{align*} \text {integral}& = \int \left (a d+a e x^2+c d x^4+c e x^6\right ) \, dx \\ & = a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \]

[In]

Integrate[(d + e*x^2)*(a + c*x^4),x]

[Out]

a*d*x + (a*e*x^3)/3 + (c*d*x^5)/5 + (c*e*x^7)/7

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
gosper	\(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\)	\(27\)
default	\(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\)	\(27\)
norman	\(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\)	\(27\)
risch	\(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\)	\(27\)
parallelrisch	\(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\)	\(27\)

[In]

int((e*x^2+d)*(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

a*d*x+1/3*a*e*x^3+1/5*c*d*x^5+1/7*c*e*x^7

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=\frac {1}{7} \, c e x^{7} + \frac {1}{5} \, c d x^{5} + \frac {1}{3} \, a e x^{3} + a d x \]

[In]

integrate((e*x^2+d)*(c*x^4+a),x, algorithm="fricas")

[Out]

1/7*c*e*x^7 + 1/5*c*d*x^5 + 1/3*a*e*x^3 + a*d*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=a d x + \frac {a e x^{3}}{3} + \frac {c d x^{5}}{5} + \frac {c e x^{7}}{7} \]

[In]

integrate((e*x**2+d)*(c*x**4+a),x)

[Out]

a*d*x + a*e*x**3/3 + c*d*x**5/5 + c*e*x**7/7

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=\frac {1}{7} \, c e x^{7} + \frac {1}{5} \, c d x^{5} + \frac {1}{3} \, a e x^{3} + a d x \]

[In]

integrate((e*x^2+d)*(c*x^4+a),x, algorithm="maxima")

[Out]

1/7*c*e*x^7 + 1/5*c*d*x^5 + 1/3*a*e*x^3 + a*d*x

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=\frac {1}{7} \, c e x^{7} + \frac {1}{5} \, c d x^{5} + \frac {1}{3} \, a e x^{3} + a d x \]

[In]

integrate((e*x^2+d)*(c*x^4+a),x, algorithm="giac")

[Out]

1/7*c*e*x^7 + 1/5*c*d*x^5 + 1/3*a*e*x^3 + a*d*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx=\frac {c\,e\,x^7}{7}+\frac {c\,d\,x^5}{5}+\frac {a\,e\,x^3}{3}+a\,d\,x \]

[In]

int((a + c*x^4)*(d + e*x^2),x)

[Out]

a*d*x + (a*e*x^3)/3 + (c*d*x^5)/5 + (c*e*x^7)/7